1. Centre for Market and Public Organisation School Markets & Correlated Random Effects Paul Clarke (with Fiona Steele, Harvey Goldstein and Rich Harris) 4 th ESRC Research Methods Festival Oxford, 5-8 July 2010

2. Outline School markets: competition & sorting Impact on multilevel modelling Methodology: correlated random effects ALSPAC data analysis Further work

3. School Markets Britain after 1944 –Local Education Authority (LEA) control –‘Catchment area’-based pupil allocation Education Reform Act (1988) –Reduced influence of LEA/catchment area –‘Quasi-market’ = ‘parental choice’ –Performance tables (GCSE, Key-stage, etc.)

4. 2-level Random Intercepts Model Standard notation Drop j to emphasize selection mechanism

5. Interpretation Ideally interpret as ‘school effects’ –e.g. teachers, ethos, size, special needs provision

6. Standard Assumptions School effects distribution No competition –Schools set s j independently (e.g. nationally) School competition –e.g. For successive cohorts 1999 and 2000 :

7. Selection/Sorting Parents’ choice of school non-random Determined by selection mechanism i.e. multinomial

8. ‘Random Effects’ Assumption Ideally school residual = school effect But only under this condition What if selection depends on school effects?

9. Impact of Selection Under weak assumptions 1 If selection independent of u then –i.e. r. effects assumption & uncorrelated 1 Schools respond to drivers of selection but population itself remains fixed

10. If selecting school j depends on u j then –i.e. r. effects assumption fails –Heteroskedastic but uncorrelated residuals Otherwise:  plus correlated residuals Impact of Selection (cont.)

11. MCMC Methodology From Browne & Goldstein (2010) 1 –Adaptive Gibbs/Metropolis-Hastings hybrid Level-two covariance matrices of form: 1 “MCMC sampling for a random intercepts model with non-independent residuals within & between cluster units”, J. Educational & Behavioral Statistics (in press)

12. ALSPAC Application Avon Longitudinal Study of Parents & Children –Followed up all births in Avon 1991-1992 –14000 children followed up Analyse primary schools (key-stage 2) –Children tested 10-11y –Mathematics and English test scores

13. Correlation Model Link function is tanh –1 Core catchment areas (CCAs) –‘Distance’ is proportional CCA overlap 1 –School’s CCA is area containing 50% of its pupils –Zero overlap  zero correlation 1 Harris & Johnston (2008), “Primary schools, markets and choice: studying polarization and the Core Catchment Areas of Schools”, Appl. Spatial Analysis 1, 59-84.

14. Results 1 Piecewise for percentiles: 10%  1 ; 10-50%  1 +  2 ; 50-90%  1 +  2 ; 90-100%  1 +  3

15. Further Work ALSPAC example: –No evidence of correlation in primary schools –Robustness to CCA definition –Analyse secondary schools Possible that –Two sources cancel out? –Possible that markets entrench difference

16. Market Dynamics School selection/sorting –Parents select schools for their children –Schools ‘select’ children School competition –Management policies implemented –Responding to other schools –Target factors influencing selection Both interrelated

17. Usual random effects structure: Not if school competition present Impact of Competition Block diagonal

18. Yes: e.g. if spatial selection element Plausibility i ukuk ujuj ulul i ujuj AB